[petsc-users] Implementing a homotopy solver

Matthew Knepley knepley at gmail.com
Fri Jul 6 13:28:32 CDT 2018

On Fri, Jul 6, 2018 at 11:44 AM Lawrence Mitchell <
lawrence.mitchell at imperial.ac.uk> wrote:

> > On 6 Jul 2018, at 17:30, zakaryah <zakaryah at gmail.com> wrote:
> >
> > ​Thanks for your help, Barry.
> >
> > I agree about the preconditioning.  I still don't understand why I don't
> need a particular solver for my shell matrix.  My reasoning is that KSP is
> easy with M but difficult with A, since A has a dense row and column,
> whereas M is entirely sparse.  Sherman-Morrison seems to be an efficient
> way of dealing with this but I could be wrong.​
> As Barry says, low-rank perturbations are not terrible for the convergence
> of Krylov methods.  E.g.
> Theorem 2.1 of https://arxiv.org/pdf/1508.07633.pdf

While I agree with you guys that Krylov methods will probably work fine, my
understanding of Strakos' results
are that theorems of this kind crumble under rounding error perturbations.
Maybe Patrick has a rejoinder for this.



> and a refinement in https://arxiv.org/pdf/1612.08369.pdf with improved
> bounds.
> Cheers,
> Lawrence

What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

https://www.cse.buffalo.edu/~knepley/ <http://www.caam.rice.edu/~mk51/>
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